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Skiplists are data-types, which can replace balanced search-trees. They are invented by Pugh. The idea is as follows:
Contrary to listnodes, which are pairs of an item and a next pointer, skipnodes are pairs of an item and a vector of next pointers. The length' of these vectors depend on each skipnode itself. They are choosen randomly in such a way, that, in the average, the number of nodes with at least k links is half the number of links with at least k-1 links, for k>1. Following the next pointers at a fixed link-level, k say, one skips all nodes with less than k pointers.
Inserting an item into a skiplist now works as follows. First one packages the item into a skipnode, where the number of links is generated randomly as described above. Second, one follows the skiplist along the highest occupied number of links as long as the skiplist's nodes point to items less then the item of the node to be inserted. Third, one steps down one level and continues following the skiplist's nodes at this new smaller level. Repeating this process until level 0 is reached we eventually find the place where our new node is to be inserted.
Some additional remarks are in order.
We described the process with a width of two, i.e. at each level in the average one node of the level below is skipped. A higher value than two for the width is possible as well, trading performance against space.
We have to decide, what to do with duplicates. We choose the following approach: The skiplist itself stores a list of either one or several numerical comparison operators. Only if the last of those operators is the special comparison operator dups (which returns constantly 0, i.e. it compares nothing) duplicates are allowed. Moreover, we arrage matters in such a way, that all nodes of duplicates with the same key have the same height, so that a search for the item which was inserted last will be found first.
In this implementation skiplists are implemented in two modules, %skiplists and skiplists. They both have the same interface. The former implements all routines without any checks, the latter imports the former with routines prefixed with % and calling those routines with checked arguments. This way you can trade security against speed ...
<procedure>(skiplists) <procedure>(skiplists sym)
documentation procedure. The first call shows the list of exported symbols, the second documentation of symbol sym.
- skiplist width max-height item? order #!rest ordersprocedure
- skiplist max-height item? order #!rest ordersprocedure
- skiplist item? order #!rest ordersprocedure
constructors: width is the jump width, max-height the maximum allowed length of pointers of an item, item? checks
- skiplist? xprprocedure
- skiplist->list slsprocedure
- skiplist->list sls levelprocedure
the list of items stored in each level
- sl-null? slsprocedure
is skiplist empty?
- sl-dups? slsprocedure
are duplicates allowed?
- sl-item? slsprocedure
item type predicate
- dups x yprocedure
trivial numerical comparison operator to be used as last order to allow duplicates.
- sl-compare slsprocedure
combined comparison function
- sl-count slsprocedure
number of items stored in skiplist
- sl-found slsprocedure
list of found items, to be called after search!
- sl-found? sls itemprocedure
is item found?
- sl-height slsprocedure
actual maximal height of nodes (can be changed)
- sl-max-height slsprocedure
absolute maximum heigth of nodes in skiplist (not changeble)
- sl-max slsprocedure
biggest item stored in skiplist
- sl-min slsprocedure
smallest item stored in skiplist
- sl-orders slsprocedure
list of orders defined in the constructor
- sl-search-level slsprocedure
down to which level a previous search descended?
- sl-width slsprocedure
width skipped on average at each search level supplied by constructor
- sl-map fn slsprocedure
- sl-map fn sls order #!rest ordersprocedure
- sl-map fn sls widthprocedure
- sl-map fn sls width order #!rest ordersprocedure
depending on the mapping function, different order procedures might be necessary
- sl-for-each proc slsprocedure
apply proc to each item in skiplist
- sl-filter ok? slsprocedure
- sl-reorder sls order #!rest ordersprocedure
- sl-restructure sls width max-heightprocedure
- sl-insert! sls item #!rest itemsprocedure
insert items into skiplist
- sl-remove! sls item #!rest itemsprocedure
remove items from skiplist
- sl-search! sls itemprocedure
searching for an item changes internal cursor transparently
- sl-clear! slsprocedure
(import skiplists) A skiplist with primary and secondary search order, allowing duplicates ;; some constructors (define sls1 (skiplist 15 fixnum? -)) (sl-width sls1) ;-> 2 (sl-max-height sls1) ;-> 15 (sl-dups? sls1) ;-> #f (define sls2 (skiplist 4 20 fixnum? - dups)) (fx= (sl-width sls2) ;-> 4 (fx= (sl-max-height sls2) ;-> 20 (sl-dups? sls2) ;-> #t ;; create ... (define item-type (lambda (x) (and ((list-of? integer?) x) (> (length x) 2)))) (define primary-order (lambda (x y) (- (car x) (car y)))) (define secondary-order (lambda (x y) (- (cadr x) (cadr y)))) (define sls3 (skiplist 3 15 item-type primary-order secondary-order dups)) ;; and populate ... (define lst1 (let loop ((k 0) (lst '())) (if (= k 100) lst (loop (+ k 1) (cons (pseudo-random-integer 10) lst))))) (define lst2 (let loop ((k 0) (lst '())) (if (= k 100) lst (loop (+ k 1) (cons (pseudo-random-integer 10) lst))))) (define lst3 (let loop ((k 0) (lst '())) (if (= k 100) lst (loop (+ k 1) (cons (pseudo-random-integer 100) lst))))) (apply sl-insert! sls3 (map (lambda (x y z) (list x y z)) lst1 lst2 lst3)) (sl-count sls3) ;-> 100 (sl-width sls3) ;-> 3 ;; inserting item and removing all with same key ... ((sl-item? sls3) '(1 2 3)) ;-> #t (sl-search! sls3 '(1 2 3)) (if (sl-found? sls3 '(1 2 3)) (apply sl-remove! sls3 (sl-found sls3))) (sl-insert! sls3 '(1 2 3)) (sl-search! sls3 '(1 2 3)) (member '(1 2 3) (sl-found sls3)) (apply sl-remove! sls3 (sl-found sls3)) (sl-search! sls3 '(1 2 3)) (null? (sl-found sls3)) ;; produce duplicates at the ends ... (sl-insert! sls3 '(-1 2 3) '(-1 2 3 4)) (sl-min sls3) ;-> '((-1 2 3 4) (-1 2 3)) (sl-insert! sls3 '(10 1 2) '(10 1 2 3) '(10 1 3)) (sl-found sls3) ;-> '((10 1 3) (10 1 2 3) (10 1 2)) (sl-max sls3) ;-> '((10 1 3) (10 1 2 3) (10 1 2)) ;; and remove them again ... (sl-search! sls3 '(-1 2 3 4)) (apply sl-remove! sls3 (sl-found sls3)) (sl-search! sls3 '(-1 2 3 4)) (null? (sl-found sls3)) ;-> #t (sl-search! sls3 '(10 1 3)) (apply sl-remove! sls3 (sl-found sls3)) (null? (sl-found sls3)) ;-> #t ;; reorder removing all dups ... (apply <= (map car (skiplist->list (sl-reorder sls3 primary-order secondary-order)))) (<= (sl-count (sl-reorder sls3 primary-order secondary-order)) (sl-count sls3)) reorder using only secondary order ... (apply < (map cadr (skiplist->list (sl-reorder sls3 secondary-order)))) (>= 10 (sl-count (sl-reorder sls3 secondary-order))) ;; restructure ... (equal? (skiplist->list sls3) (skiplist->list (sl-restructure sls3 2 10))) ;; filter ... ((list-of? odd?) (map caddr (skiplist->list (sl-filter sls3 (lambda (x) (odd? (caddr x))))))) ;; map ... (let ((fn (lambda (x) (* 2 x)))) (equal? (map fn (skiplist->list sls3)) (skiplist->list (sl-map sls3 fn))))
Mar 25, 2019
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- port from chicken-4